Math Problem Statement
Solution
The function provided in the image is:
To determine where is continuous, we need to identify the values of that cause the denominator to be zero since these will be the points of discontinuity.
The denominator is zero when:
Solving this equation, we find:
Therefore, the function is discontinuous at and . Elsewhere, the function is continuous since the numerator and denominator are polynomials, and polynomials are continuous everywhere.
So, the function is continuous for all except at and .
Would you like to explore how the limits behave at these points or have other questions on continuity or limits?
Here are five related questions you might consider:
- What is the behavior of as approaches 5?
- How does behave as approaches -4?
- Can be redefined at or to make it continuous at these points?
- What is the limit of as approaches infinity?
- How would the graph of look around the points of discontinuity?
Tip: When checking for continuity of a rational function, it’s helpful to factorize the denominator and look for common factors in the numerator, which can simplify the expression and potentially remove some points of discontinuity.
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Math Problem Analysis
Mathematical Concepts
Continuity
Rational Functions
Discontinuity
Formulas
f(x) = (x - 7) / ((x - 5)(x + 4))
Theorems
The Continuity Theorem for Rational Functions
Suitable Grade Level
Grades 11-12
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